Question: The differentiable functions $x$ and $y$ are related by the following equation: $\sin(y)=-5x$ Also, $\dfrac{dy}{dt}=10$. Find $\dfrac{dx}{dt}$ when $y=-\pi$.
Let's start by differentiating the equation $\sin(y)=-5x$ with respect to $t$. $\begin{aligned} \sin(y)&=-5x \\\\ \cos(y)\cdot\dfrac{dy}{dt}&=-5\cdot\dfrac{dx}{dt} \end{aligned}$ We are given that $\dfrac{dy}{dt}=10$, and we want to find $\dfrac{dx}{dt}$ when $y=-\pi$. Let's plug ${y=-\pi}$ and ${\dfrac{dy}{dt}=10}$ into the equation we obtained: $\begin{aligned} \cos({y})\cdot{\dfrac{dy}{dt}}&=-5\cdot\dfrac{dx}{dt} \\\\ \cos({-\pi})\cdot{10}&=-5\cdot\dfrac{dx}{dt} \\\\ -10&=-5\cdot\dfrac{dx}{dt} \\\\ 2&=\dfrac{dx}{dt} \end{aligned}$ In conclusion, when $y=-\pi$, the value of $\dfrac{dx}{dt}$ is $2$.